# Global existence of solution of ODE

I have to say if following statements are true or false. For the initial value problem: $$x'=f(x), x(t_0)=x_0$$ with $$f:\mathbb{R^n}\rightarrow\mathbb{R^n}$$ , $$t_0\in\mathbb{R},x_0\in\mathbb{R^n}$$

1. If $$f$$ is continuous and bounded then for all $$t_0\in\mathbb{R},x_0\in\mathbb{R^n}$$ there exists a global solution.
1. If $$f$$ is a linear function then for all $$t_0\in\mathbb{R},x_0\in\mathbb{R^n}$$ there exists a unique global solution.

I have a small problem with the concept of a global solution. I know from the Peano Theorem that if $$f$$ is continuous then there exists a local solution, if $$f$$ is locally Lipschitz continuous, then this solution is unique(not sure if locally or everywhere). Then I think that if we know that it is locally Lipschitz and continuous we can expand the solution to the maximum existence interval. And if this interval is the whole domain then we can say the solution is global. Is this correct? What are other things I have to know so that I can say if it has a global solution or not and especially if this solution is unique?

Now if the function $$f$$ is linearly bounded, one finds that the solutions are exponentially bounded, so a divergence to space infinity is not possible in finite time. It remains that the time coordinate extends to infinity.